Hello. I'm quite new to calculus and frankly speaking I'm an autodidact. Could you tell me how to solve this:

$\displaystyle f: [\frac{\pi}{2}, \pi) \ni x \rightarrow \frac{1}{sinx} \in R$

Prove that there exists a inverese of $\displaystyle f$. In what set is $\displaystyle f^{-1}$ differentiable? Calculate $\displaystyle f^{-1'}$

I would really appreciate a thorough explanation. Thank you.